Dom-
Is it possible that the maps/apps that are being shared contain a layer that Esri classifies as Premium Content? Or is there a layer in the map that is not publicly shared? Esri has certain data that requires an organization login. It can be any organization login--it doesn't have to be the organization that created the map. For example, if I was using the LearnGIS account and created a map with certain demographic data, I could share it with everyone. If I then signed out and attempted to access the map, it would request a sign in. I could sign in with my school account and the map would appear. Premium content just requires you to "prove" you are a member of an organization.
In the second situation, suppose the "LearnGIS" organization had created a layer and shared it with the LearnGIS organization only. I could create a map or app that contained the layer, share the map/app with everyone, and then sign out. Now when I tried to sign in with my school account, I still wouldn't be able to access the map, because the layer was only shared with the LearnGIS organization. As I'm not an administrator in the "LearnGIS" organization, I wouldn't be able to share that map layer with everyone, so there wouldn't be an easy solution, unless I could somehow make a copy of that layer and share it with everyone.
Esri attempts to warn you when you attempt to share a map with a layer that is not publicly shared, but it doesn't always seem to catch it, and seems to warn you less often when sharing an app.
For publishers/administrators in an organization there is a solution for Esri's Premium Content. It's described in an Esri blog at: http://blogs.esri.com/esri/arcgis/2015/05/07/sharing-arcgis-online-demographic-maps-with-the-public/
Hope this helps.
Ina
